Problem: Simplify; express your answer in exponential form. Assume $t\neq 0, z\neq 0$. $\dfrac{{(t^{-3}z^{5})^{-1}}}{{(t^{-5}z^{2})^{2}}}$
To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(t^{-3}z^{5})^{-1} = (t^{-3})^{-1}(z^{5})^{-1}}$ On the left, we have ${t^{-3}}$ to the exponent ${-1}$ . Now ${-3 \times -1 = 3}$ , so ${(t^{-3})^{-1} = t^{3}}$ Apply the ideas above to simplify the equation. $\dfrac{{(t^{-3}z^{5})^{-1}}}{{(t^{-5}z^{2})^{2}}} = \dfrac{{t^{3}z^{-5}}}{{t^{-10}z^{4}}}$ Break up the equation by variable and simplify. $\dfrac{{t^{3}z^{-5}}}{{t^{-10}z^{4}}} = \dfrac{{t^{3}}}{{t^{-10}}} \cdot \dfrac{{z^{-5}}}{{z^{4}}} = t^{{3} - {(-10)}} \cdot z^{{-5} - {4}} = t^{13}z^{-9}$